The Fourier series of the third integral of motion

The coefficients of the Fourier series of the third integral of motion, [GRAPHICS] in which f denotes the direction of the velocity in the meridional plane of the potential Phi(R, z) comoving with the object, can be represented in the form [GRAPHICS] The quantities C-si and D-si(R, z) are interrelated by a system of recurrence differential equations flowing from the Boltzmann equation. If we know the potential Phi(R, z) and the function C-0, which has the meaning of a partial density in the Poisson equation, all subsequent coefficients C-si and D-si in this system are determined by differentiation and algebraic operations.